The PIC has the ultimate responsibility per the FARs, but that simply means s/he has a legal standign to refuse the aircraft for the flight. That does NOT in ANY way, shape, or form, authorize a pilot to give orders to anyone. That DOES, however, allow the pilot to refuse the flight or aircraft, at which point s/he will have to justify it to their superiors. If the company disagrees, well, then there's an FAR battle. The FOM, CSM, GOM, and all the other nifty manuals we have are ALL approved by the FAA as a condition of our Part 121 certificate. I'm reasonably certain that changes either need to be FAA approved or fall within some type of FAA scope for change.
If an aircraft is hot ground air should be requested by the PIC. If the PIC (or SIC or inflights for that matter) request PC air is to be connected. As far as the PIC goes that's policy, with the others it's just common sense. If PC air is not available, or not sufficient, the PIC (or SIC) should take action to start the APU and use the pakcs to cool the aircraft.
Why is this such an issue? Have air and the pilot wants ot? GIVE IT TO HIM! The rampers tell the pilot there is no air to be had? START THE APU! Simple. Done. Make it happen!
EDIT: Forgot to add that there WILL be a point that the packs can NOT keep up with the heat on some aircraft. These aircraft should be priorities for PC air.